Warm, comforting, and deeply satisfying, this Korean rice porridge is perfect for when you're sickāand when you're not.
In a medium bowl, cover rice with at least 2 inches cold water and soak for 1 hour. Drain well.
Meanwhile, in a large Dutch oven or similarly sized pot, combine chicken, whole garlic cloves, ginger, and scallion whites with 2 quarts (2L) cold water. Bring to a simmer, then cover and cook at a gentle simmer until thickest part of breast close to bone registers 150°F (66°C) on an instant-read thermometer and joint between thighs and body registers at least 175°F (80°C), about 45 minutes.
Carefully transfer chicken to work surface and let stand until cool enough to handle. Set a fine-mesh strainer over a medium heatproof bowl and strain broth through it; discard solids.
In a 3-quart saucier or saucepan, heat sesame oil over medium heat until shimmering. Add drained rice and cook, stirring frequently, until a sticky film develops on bottom of pot, about 3 minutes. Add carrot, onion, and zucchini and cook, stirring frequently, until vegetables have softened slightly and rice grains are slightly translucent around their edges, about 2 minutes.
Add 6 cups (1.4L) strained broth and stir to scrape up any starchy coating on bottom of pan. Bring to a rapid simmer, then cook, stirring occasionally to make sure nothing sticks to bottom of pan, until rice grains are plump and tender and liquid is thickened, about 30 minutes.
Meanwhile, shred chicken meat, discarding skin and bones.
Add two-thirds of both the chicken meat and scallion greens to juk along with remaining 2 cloves minced garlic and simmer until chicken is warmed through; if juk is thicker than you'd like, thin as desired with remaining broth (or reserve extra broth for another use). Season juk with salt to taste.
Ladle juk into individual serving bowls, garnish with remaining shredded chicken and scallion greens, sprinkle with sesame seeds, and serve.